Banach?Tarski paradox

First stated by Stefan Banach and Alfred Tarski in 1924, the Banach?Tarski paradox or Hausdorff?Banach?Tarski paradox is the famous "doubling the ball" paradox,

Formally

Let A and B be two subsets of Euclidean space. We call them equi-decomposable if they can be represented as finite unions of disjoint subsets A=cup_{i=1}^n A_i and B=cup_{i=1}^n B_i such that, for any i, the subset A_i is congruent to B_i. Then, the paradox can be reformulated as follows:

Related Topics:
Euclidean space - Unions - Disjoint - Congruent

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:The ball is equi-decomposable with two copies of itself.

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For the ball, five pieces are sufficient to do this; it cannot be done with fewer than five. There is an even stronger version of the paradox:

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:Any two bounded subsets of 3-dimensional Euclidean space with non-empty interior are equi-decomposable.

Related Topics:
Subset - Euclidean space - Empty - Interior

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In other words, a marble could be cut up into finitely many pieces and reassembled into a planet, or a telephone could be transformed into a water lily. These transformations are not possible with real objects made of a finite number atoms and bounded volumes, but it is possible with their geometric shapes. The Banach?Tarski paradox is made somewhat less bizarre by pointing out that there is always a function that can map one-to-one the points in one shape to another. For example, two balls can be transformed bijectively to a similarly infinite subset of itself (such as one ball). Likewise, we can make one ball into a large or smaller ball by simply multiplying the radius of each point in the ball, using spherical coordinates, by a constant. However, such transformations in general are non-isometric or involve an uncountably infinite number of pieces. The surprising consequence of the Banach?Tarski paradox is that it can be done with only rotation and translation (isometric mapping) of a finite number of pieces.

Related Topics:
Atoms - Bijectively - Spherical coordinates - Uncountably

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What makes the paradox possible is that the pieces are infinitely convoluted. Technically, they are not measurable, and so they do not have "reasonable" boundaries or a "volume" in the ordinary sense. It is impossible to carry out such a disassembly physically because disassembly "with a knife" can create only measurable sets. This pure existence statement in mathematics points out that there are many more sets than just the measurable sets familiar to most people.

Related Topics:
Measurable - Boundaries

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The paradox also holds in all dimensions higher than three. It does not hold for subsets of the Euclidean plane. (The statement above does not apply to a two-dimensional subset of three-dimensional space, since such a subset would have empty interior.) Still, there are some paradoxical decompositions in the plane: a disc can be cut into finitely many pieces and reassembled to form a solid square of equal area; see Tarski's circle-squaring problem.

Related Topics:
Euclidean plane - Disc - Tarski's circle-squaring problem

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The paradox shows that it is impossible to define "volume" on all bounded subsets of Euclidean space such that equi-decomposable sets will have equal volume.

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The proof is based on the earlier work of Felix Hausdorff, who found a closely related paradox 10 years earlier; in fact, the Banach?Tarski paradox is a simple corollary of the technique developed by Hausdorff.

Related Topics:
Felix Hausdorff - Closely related paradox

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Logicians most often use the term "paradox" for a statement in logic which creates problems because it causes contradictions, such as the Liar paradox or Russell's paradox. The Banach?Tarski paradox is not a paradox in this sense but rather a proven theorem; it is a paradox only in the sense of being counter-intuitive. Because its proof prominently uses the axiom of choice, this counter-intuitive conclusion has been presented as an argument against adoption of that axiom.

Related Topics:
Liar paradox - Russell's paradox - Axiom of choice

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